Causal models for qualitative and mixed methods inference

Mixed methods

Macartan Humphreys and Alan Jacobs

1 Population parameters

1.1 Population-level causal questions

Average causal effects for a population
- e.g., What is the average effect of \(X\) on \(Y\)
Proportion of different effects in a population
- What share of cases in the population have positive effects?
- What share have negative effects?
Causal pathways
- e.g., How commonly does \(X\) affect \(Y\) through \(M\) (vs. \(N\)) in the population?

1.2 Causal queries on a DAG: population-level average causal effect

What is the average effect of \(X\) on \(Y\) in the population?
- This is a question about the values in \(\lambda^Y\)

1.3 Causal queries on a DAG: population-level average causal effect

With binary variables, the average effect is always the difference between
- The share of cases with a positive effect
- The share of cases with a negative effect
\(\lambda_{01} - \lambda_{10}\)

1.4 Causal queries on a DAG: population-level average causal effect

\(ATE = \lambda^Y_{01} - \lambda^Y_{10}\)

1.5 ATE with a mediator

We need to

Combine two ways of generating positive effects
Combine two ways of generating negative effects
Subtract the first from the second

1.6 ATE with a mediator

Two ways of generating positive effects: \(\lambda^M_{01} \times \lambda^Y_{01}\) + \(\lambda^M_{10} \times \lambda^Y_{10}\)
Two ways of generating negative effects: \(\lambda^M_{10} \times \lambda^Y_{01}\) + \(\lambda^M_{01} \times \lambda^Y_{10}\)
Subtract the first from the second

1.7 Causal queries on a DAG: how often does \(X\) have a positive effect on \(Y\)?

Proportion of cases with positive effect = \(\lambda^Y_{01}\)

1.8 Causal queries on a DAG: how often does \(X\) have a positive effect on \(Y\)?

Proportion of cases with positive effect = \(\lambda^M_{01} \times \lambda^Y_{01}\) + \(\lambda^M_{10} \times \lambda^Y_{10}\)

1.9 Causal queries on a DAG: pathway questions

We can also pose the pathway query at population level
- What is the share of cases for which \(X\) has a positive effect on \(Y\) through \(M\)?
A question about joint \(\lambda^M\) and \(\lambda^Y\) distributions

1.10 How will we answer questions about populations?

We need to learn about those \(\lambda\)’s
- About the proportions of the population with different kinds of causal effects
We will have a prior belief about those proportions
When we see data on lots of cases, we will update those beliefs about proportions
- From a prior distribution over \(\lambda\) to a posterior distribution over \(\lambda\)

2 Intuition

2.1 How do we “update” our models?

We’ve talked about process tracing a single case to answer a case-level query
- Here the model is fixed
- We use the model + case data to answer questions about the case
We can also use data to “update” our models
- Use data on many cases to learn about causal effects in the population
Allows mixing methods: using data on lots of cases, we can learn about probative value of process-tracing evidence
The core logic: we learn by updating population-level causal beliefs toward beliefs more consistent with the data

2.2 Start with a DAG

2.3 Large-\(N\) estimation of \(ATE\): what happens to beliefs over parameters

We collect data on \(I\), \(M\), and \(D\) for a large number of cases
We observe a strong positive correlation
We will think there’s a positive average effect

2.4 Large-\(N\) estimation of \(ATE\): what happens to beliefs over parameters

Now, we will update on both \(\lambda^I\) , \(\lambda^M\) and \(\lambda^D\)
An \(I \rightarrow D\) effect can only happen if \(I\) affects \(M\) and \(M\) affects \(D\), in specific ways
Two possible combinations of effects can generate a positive \(I \rightarrow D\) effect
- \(I \rightarrow M\) is positive, \(M \rightarrow D\) is positive
- \(I \rightarrow M\) is negative, \(M \rightarrow D\) is negative
So we will come to put more weight on a joint distribution of \(\lambda^M\) and \(\lambda^D\) in which there are lots of cases with one of these two combinations
…and less posterior weight on all other combinations of effects

2.5 General procedure

Key insight:

If we suppose a given set of parameter values we can figure out the likelihood of the data given those values.
We can do this for all possible parameter values and see which ones are more in line with the data

That, with priors, is enough to update:

\[p(\lambda | D) = \frac{p(D | \lambda)p(\lambda)}{p(D)}\]

2.6 Learning with confounding

Say we just observe a positive Inequality-Democratization correlation
Could be because Inequality causes Democratization
Could be because of confounding

2.7 Learning with confounding

So observing \(M\) helps!
- Allows us to learn if \(I\) affects \(M\)
- And if \(M\) affects \(D\)
Suppose we see:
- a strong positive \(I\) to \(M\) correlation
- a strong positive \(M\) to \(D\) correlation
We update toward belief that \(I\) to \(D\) effect is common, and that confounding is uncommon
If \(I/M\) and \(M/D\) correlations are weak/absent, then confounding likely
Process data helps address the deep problem of confounding

2.8 In sum: learning from data

For any data pattern, we gain confience in parameter values more consistent with the data
For single-case inference, we must bring background beliefs about population-level causal effects
For multiple cases, we can learn about effects from the data
Large-\(N\) data can thus provide probative value for small-\(N\) process-tracing
All inference is conditional on the model

3 Mixed methods

3.1 A DAG

We’ll want to learn about the \(\theta\)’s and the \(\lambda\)’s
We need to observe nodes to learn about other nodes
We can potentially observe 3 nodes here: \(X, M\), and \(Y\)

3.2 A typical “quantitative” data structure

Data on exogenous variables and a key outcome for many cases
E.g., data on inequality (\(I\)) and democracy (\(D\)) for many cases

3.3 A typical “qualitative” data structure

Data on exogenous variables and a key outcome plus elements of process for a small number of cases
- Finite resources mean tradeoffs between extensive and intensive data collection
E.g., data on inequality (\(I\)), mass mobilization (\(M\)), and democracy (\(D\)) for many cases

3.4 Mixing qualitative and quantitative

What if we combine extensive data on many cases with intensive data on a few cases?

A non-rectangular data structure

3.5 Non-rectangular data

A data structure that neither standard quantitative nor standard qualitative approaches can handle in a systematic way
Not a problem for the Integrated Inferences approach
We simply ask:
- Which causal effects in the population are most and least consistent with the data pattern we observe?
That is, what distribution of causal effects in the population, for each node, are most consistent with this data pattern?
CausalQueries uses information wherever it finds it

3.6 Mixing in practice

For Bayesian approaches this mixing is not hard.

Critically though we maintain the assumption that cases for “in depth” analysis are chose at random—otherwise we have to account for selection processes.

What is the probability of seeing these two cases:

\(X=1, M = 1, Y = 1\) case
\(X=1, Y=1\) (no data on \(M\))

given parameters \(\lambda\):

3.7 Mixing in practice

The probability of 1 is:

\[p_{111}= \lambda^X_1 \times (\lambda^M_{01} + \lambda^M_{11}) \times (\lambda^Y_{01} +\lambda^Y_{11})\]

The probability of 2 is:

\[p_{1?1} = \lambda^X_1\times \left((\lambda^M_{01} + \lambda^M_{11}) \times (\lambda^Y_{01} +\lambda^Y_{11}) + (\lambda^M_{10} + \lambda^M_{00}) \times (\lambda^Y_{10} +\lambda^Y_{11}) \right)\]

So the probability of this data is just:

\[p(D|\lambda) = p_{111} * p{1?1}\]

3.8 Mixing in practice

Insight:

If we imagine possible parameter values we can figure out the likeihood of any data type – quantitative, qualitative or mixed.